Question

the annual salaries in a company are approximately normally distributed with a mean of r50000 and a standard deviation of r20000. what what is the probability of people who earn between r45000 and r65000?

Answer 1

Let X be random variable representing the annual salaries in a company.

X is normally distributed with a mean $\mu = 50000$ and a standard deviation $\sigma = 20000$. Therefore, $X = \sigma Y +\mu$ where random variable Y has a standard Gaussian distribution with the cumulative distribution function $P(Y<y) =\Phi(y) = \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{y}e^{-t^2/2}dt$

$P(x_1\leq X\leq x_2) = P(x_1\leq \sigma Y +\mu\leq x_2) = P(\frac{x_1-\mu}{\sigma} \leq Y \leq \frac{x_2-\mu}{\sigma}) = \Phi(y_2) - \Phi(y_1)$

where

$y_1 = (x_1-\mu)/\sigma = (45000 - 50000)/20000 = -0.25$

$y_2 = (x_2-\mu)/\sigma = (65000 - 50000)/20000 = 0.75$

$P(x_1\leq X\leq x_2) = \Phi(0.75) - \Phi(-0.25)= 0.7734 - 0.4013 = 0.3721$

Answer. The probability of people who earn between r45000 and r65000 is 0.3721